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# 10 into 139c e and evaluate the integrals let us

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Unformatted text preview: ; (1.3.11a) where hj = xj ; xj Thus, U (x) is constant on xj ; j = 1 2 : : : N: 1 (1.3.11b) xj ] and is given by the rst divided di erence U (x) = cj ; cj 1 x 2 xj 1 xj ]: hj Substituting (1.3.11) and a similar expression for V (x) into (1.3.9b) yields Zx S (V U ) = Aj p dj 1 dj ] ;1=hj ;1=hj 1=hj ] cjc 1 dx 1=hj j x ;1 0 1 ; ; 0 ; 0 j ; ; j or AS (V U ) = dj 1 dj ] j ; Z xj xj;1 ! 1=hj p ;1=h2 ;1=hj2 dx cjc 1 : 1=hj j j 2 2 ; The integrand is constant and can be evaluated to yield p Kj = h AS (V U ) = dj 1 dj ]Kj cjc 1 j j ; ; j 1 ;1 ;1 1: (1.3.12) The 2 2 matrix Kj is called the element sti ness matrix. It depends on j through hj , but would also have such dependence if p varied with x. The key observation is that Kj can be evaluated without knowing cj 1, cj , dj 1, or dj and this greatly simpli es the automation of the nite element method. The evaluation of AM proceeds similarly by substituting (1.3.10) into (1.3.9d) to j obtain Zx M (V U ) = Aj q dj 1 dj ] j 1 j 1 j ] cjc 1 dx: ; ; j ; xj;1 j ; ; ; j With q a constant, the integrand is a quadratic polynomial in x that ma...
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